Bifurcations in flow patterns: Some applications of the qualitative theory of differential equations in fluid dynamics

Bifurcations in Flow Patterns

Some applications of the qualitative theory of

differential equations in fluid dynamics

Proefschrift

ter verkrijging van de graad van doctor aan

de Technische Universiteit Delft, op gezag

van de Rector Magnificus, prof. dr. J.M. Dirken,

in het openbaar te verdedigen ten overstaan

van een commissie door het College van

Dekanen daartoe aangewezen,

op dinsdag 31 mei 1988 te 14.00 uur.

door

Pieter Gerrit Bakker

vliegtuigbouwkundig ingenieur

geboren te Groningen

TR diss

1632

Dit proefschrift is goedgekeurd door de promotor

Prof. dr. ir. J.W. Reyn

Abstract

In the present thesis the qualitative theory of differential equations' is used along with topological considerations to discuss problems in fluid dynamics and gasdynamics.

Special attention is given to the qualitative aspects of flow fields, in par­ ticular to the geometry, the shape and the structural stability of streamline patterns.

The theory relies much on critical point analyses and on bifurcations in vector fields. Local solutions of the flow equations are derived to discuss changes in flow topology in conjunction with bifurcations of critical points.

The theory is applied to topics of inviscid, nonlinear conical flows and of steady viscous flows over plane walls.

Contents

Page

Preface 1

Chapter I Some elements of the qualitative theory of differential equations

1. Phase space representation of a dynamical system 5

2. Phase portraits near singular points 9 3. Topological structure of phase portraits, structural stability, 13

bifurcation

4. Higher-order singularities in R2 20

5. Bifurcation of vector fields, unfoldings 27

6. Center manifolds 33 7. An approach to physical unfoldings in flow patterns 4l

8. References 43

Chapter II Topology of conical flow patterns

1. Introduction 45 1.1. Concepts and definitions 45

1.2. A survey of conical flow theory 46 1.3. Conical streamlines, conical stagnation points 48

1.4. Transition phenomena in conical flow patterns 51 2. Local conical stagnation point solutions in irrotational flow 55

2.1. Conical potential equation 55 2.2. Conical stagnation point solutions 56

3. Classification of conical singular points in conical flows 61

3.1. First-order conical stagnation points 61 3.2. Irrotational attachments and separations 66 3.3- Higher-order conical stagnation points 70 4. Analytical unfoldings in conical flows 75

4.1. Bifurcation parameters 75 4.2. Approximate solutions near regular points 76

4.3. Saddle-node bifurcation 79 4.4. Bifurcation of topological saddle point 83

4.5- Bifurcation of topological node 90 5. External corner flow; a nonanalytic unfolding of a starlike node 92

5.1. The flow around an external corner 92 5.2. Boundary conditions and bifurcation modes 95

5.3- Bifurcations of the starlike node 98 5.4. Symmetrical external corners 104 5.5- Transition of oblique saddle to starlike node 109

i i i

-Chapter III Topological aspects of steady viscous flows near plane walls

1. Local solutions of the Navier-Stokes equations

2. Steady viscous flow near a plane wall, elementary singular points 2.1. Approximate solutions near a plane wall

2.2. Elementary singular points located at the wall 2.3- Elementary singular points in the flow

3. Higher-order singularities in the flow pattern 3.1. Higher-order singular points in the flow field 3.2. Higher-order singular points on the wall

k. Unfolding of a topological saddle point of the third order 4.1. Local phase portraits of the unfolding

k.2. Incipient bubble separation 4.3. Separation along a moving wall

k.k. MRS-criterion for separation in flows along a moving wall 4.5- Unfolding model for moving wall separations

5. Unfolding of a topological saddle point of the fifth order 5.1. Description of the unfolding

5.2. Bubble capturing by a secondary separation

6. Unfolding of a saddle point with three hyperbolic sectors in a half plane

6.1. Universal physical unfolding 6.2. Bifurcation sets, flow patterns

7. Unfolding of a saddle point with two or four hyperbolic sectors in a half plane

7.1. Universal physical unfolding 7.2. Determination of codimension

7-3- Neighbouring singular points, local bifurcation sets B and B 7.4. Flow patterns and global bifurcation sets B . and B _

8. Viscous flow near a circular cylinder at low Reynolds numbers 8.1. Description of flow topology

8.2. Symmetrical bifurcations

8.3. Asymmetrical bifurcations, transition scenario's 9• References

Samenvatting

About the author

-1-Preface

The main idea of the present thesis is to demonstrate that the qualitative theory of differential equations, when applied to problems in fluid- and gas-dynamics, will contribute to the understanding of qualitative aspects of fluid flows, in particular those concerned with geometrical properties of flow fields such as shape and stability of its streamline patterns.

It is obvious that insight into the qualitative structure of flow fields is of great importance and appears as an ultimate aim of flow research. For, qualita­ tive insight fashions our knowledge; it serves as a good guide for further quantitative investigations. Moreover, qualitative information can become very useful, especially when it is applied in close correspondence with numerical methods, in order to interpret and value numerical results. A qualitative analysis may be crucial for the investigation of the flow in the neighbourhood of singularities where a numerical method is not reliable anymore due to discretisation errors being unacceptable.

Up till now, familiar research methods - frequently based on rigorous analyses, careful numerical procedures and sophisticated experimental techniques - have increased considerably our qualitative knowledge of flows, albeit that the information is often obtained indirectly by a process of a careful but cumber­ some examination of quantitative data.

In the past decade, new methods are under development that yield the qualitative information more directly.

These methods, make use of the knowledge available in the qualitative theory of differential equations and in the theory of bifurcations.

The qualitative theory of differential equations as applied to dynamical systems of two-and three-dimensional vector fields appears to be very useful, in order to determine the topology and structural stability of streamline patterns, occurring in fluid dynamics. The theory originates from the work of Poincaré (1880) and is further developed by Birkhoff (~ 1927). Lyapunov (- 19*»9). Andronov (1937) and his co-workers, Arnold (1963) and many others.

For references see Chapter I (Guckenheimer and Holmes

(1983))-In the present thesis this theory is applied to flows, which can be described by two variables x and y and the streamlines are the solution curves of a system of the second-order, -^ = P(x,y), -^ = Q(x,y), where P and Q are related to the components of the velocity.

2

-A survey of some elements of the q u a l i t a t i v e theory of differential equations i s

given in Chapter I . The review introduces and discusses in some d e t a i l the most

important c o n c e p t s , theorems, and methods t h a t are used in the subsequent

analyses; for rigorous mathematical proofs the r e a d e r w i l l be r e f e r r e d to the

l i t e r a t u r e . A c t u a l l y Chapter I may be consulted in order to be informed about

notions as phase p o r t r a i t s of a dynamical system, singular p o i n t s , t o p o l o g i c a l

s t r u c t u r e , s t r u c t u r a l s t a b i l i t y , degenerate systems, b i f u r c a t i o n of vector

f i e l d s , unfoldings, co-dimension and center-manifold theory.

The f i r s t a p p l i c a t i o n of the q u a l i t a t i v e theory in t h i s t h e s i s , concerns the

flow geometry of three-dimensional inviscid conical gas flows. I t i s t r e a t e d in

Chapter I I . Conical flows have the specific property that the velocity of the

gas p a r t i c l e s and the q u a n t i t i e s , defining the s t a t e of the gas, e . g . p r e s s u r e

density and temperature, are constant along rays originating from a common point

(conical center) . A three-dimensional c o n i c a l flow i s e s s e n t i a l l y two-dimen­

s i o n a l and i s decribed adequately on a unit sphere (around the conical center)

on which the flow geometry i s displayed by conical streamlines. Conical stream­

l i n e s are- defined as follows. For conical flows, the family of s p a t i a l stream­

lines passing the same ray form a conical stream s u r f a c e with v e r t e x in the

c o n i c a l c e n t e r . The i n t e r s e c t i o n of a c o n i c a l stream surface with the unit

sphere i s a c o n i c a l s t r e a m l i n e . Along a c o n i c a l s t r e a m l i n e the entropy i s

constant or i t jumps i f a shock i s passed.

The conical flow geometry on the u n i t sphere i s governed by a second-order

dynamical system. The corresponding vector field i s determined by the cross-flow

velocity, being the velocity component tangential to the unit sphere. In p o i n t s

on t h e u n i t sphere where t h i s component vanishes the flow is purely r a d i a l ;

these points are called conical stagnation points and appear as s i n g u l a r i t i e s of

the second-order dynamical system governing the conical streamlines. In Chapter

I I , possible flow patterns near conical stagnation points are studied and t h e i r

s t r u c t u r a l s t a b i l i t y i s examined.

Transitions to a different conical flow pattern are interpreted as b i f u r c a t i o n s

of s t r u c t u r a l l y unstable higher-order conical stagnation points.

Chapter I I gives a complete c l a s s i f i c a t i o n of f i r s t - o r d e r c o n i c a l s t a g n a t i o n

p o i n t s and s t a r t s with t h e c l a s s i f i c a t i o n of higher-order conical stagnation

points, by considering second- and t h i r d - o r d e r p o i n t s , t h e s e p o i n t s being of

most p r a c t i c a l i n t e r e s t in real flow problems.

Chapter I I ends up with a q u a l i t a t i v e examination of the i n v i s c i d flow around

t h r e e d i f f e r e n t c o n i c a l b o d i e s : circular cones, delta-wings with arrow-shaped

cross-section and external corners.

-3-In Chapter I I I the qualitative theory will be applied to steady two-dimensional

incompressible viscous flows along the surface of a plane or along a s l i g h t l y

curved w a l l . A second-order dynamical system, whose t r a j e c t o r i e s represent

streamlines, i s derived.

A s i n g u l a r - p o i n t - a n a l y s i s i s performed in order t o obtain detailed information

about the local flow topology.

In p a r t i c u l a r , s i n g u l a r points on the wall surface are of i n t e r e s t , because in

these p o i n t s the shear s t r e s s vanishes i n d i c a t i n g flow s e p a r a t i o n or flow

attachment.

As in Chapter I I the singular points are d i s t i n g u i s h e d i n t o f i r s t - o r d e r and

h i g h e r - o r d e r s i n g u l a r i t i e s , the former appear only as centerpoints and saddle

points. First-order saddle points that are located a t the wall surface represent

s o l u t i o n s that are recognized as the c l a s s i c a l Oswatitsch-Legendre solution for

flow separation or attachment. The Oswatitsch-Legendre solution i s s t r u c t u r a l l y

s t a b l e i n the sense t h a t a n a l y t i c a l p e r t u r b a t i o n s i n streamwise p r e s s u r e

gradients and/or shear s t r e s s gradients will not affect the flow topology near

the separation (attachment) point.

Apart from the s t r u c t u r a l l y stable solutions, s t r u c t u r a l l y u n s t a b l e s o l u t i o n s

appear as w e l l , they occur as higher-order or degenerate s i n g u l a r i t i e s of the

dynamical system that governs the streamline p a t t e r n .

The study of these degenerate s i n g u l a r i t i e s ; their unfoldings and bifurcational

behaviour, will be the main subject of Chapter I I I . Unfoldings, r e s u l t i n g in

d e g e n e r a t e s i n g u l a r i t i e s f a l l i n g a p a r t i n t o a number of f i r s t - o r d e r s i n ­

g u l a r i t i e s , d e s c r i b e p o s s i b l e s t r u c t u r a l l y s t a b l e flow p a t t e r n s in viscous

incompressible flow. These patterns, being more complex for s i n g u l a r i t i e s with a

higher degree of degeneracy, are interpreted physically. Some of them appear to

be very common in aerodynamics, others are new and are concerned with topics in

laminar flows as:

- genesis of laminar separation bubbles

- flow separation on moving walls (Moore-Roth-Sears c r i t e r i o n )

- interference of separations and attachments, and

- formation of asymmetric standing eddies in the near wake behind a body (vortex

shedding).

-5-Chapter I Some elements of the qualitative theory of differential equations

1. Phase space representation of a dynamical system

With the aim of easy reference we will give in this chapter some elements of the qualitative theory of dynamical systems which we will use in the next chapters. The theory of dynamical systems has been extensively studied over a long period of time. Although many questions remain as yet unsolved, a large amount of results has been obtained and is available for applications.

Numerous textbooks, articles and papers testify of the progress made in this branch of mathematics.

A comprehensive survey of important achievements, including recent developments and advanced methods, is given by Guckenheimer and Holmes (1983): 'Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields'.

This book, written with a strong view to combine pure mathematical reasoning and applicability to practice, has inspired me in writing this chapter.

The theory of dynamical systems aims to study the time behaviour of evolutionary systems which are described mathematically by an equation of the form:

where X = X(t) e R is a n-vector and f(X,t) is a sufficiently smooth function defined on some subset U Ê R x R.

In applications t is usually interpreted as time. The function f defines a vector field in the n-dimensional space the so-called phase space.

The solutions of the system, representing sequential states of the evolutionary process, appear as integral curves of the vector field in the phase space. In the literature on this subject, one encounters several names for these curves, we quote: solution curves, trajectories, (phase-)paths and orbits. In the same way, the terms phase portrait, phase pattern, flow or trajectory pattern are used to indicate the whole set of solution curves in the phase space.

Dynamical systems where f does not explicitly depend on time are called autono­ mous and the trajectories of such systems do not change if time goes on.

In the following we only need to study these autonomous systems, being expressed by the equation

x = f ( x ) , x e R

n

_(1.1)

In a c l a s s i c a l treatment of such a system, see for example Coddlngton and

Levin-son (1955). the attention i s mainly d i r e c t e d to the p r o p e r t i e s of i n d i v i d u a l

s o l u t i o n curves and culminates into questions about e x p l i c i t time behaviour and

dependency on i n i t i a l conditions of the solutions.

A d i f f e r e n t approach t o the study of dynamical systems i s obtained i f one

considers families of solution curves; then q u a l i t a t i v e questions a r i s e such as:

do t h e r e e x i s t steady and/or p e r i o d i c s o l u t i o n s , what are t h e i r s t a b i l i t y

p r o p e r t i e s .

Domains in t h e phase space where f (X) i s a nonvanishing vector function are

called regular; the phase p o r t r a i t s in a regular domain are relatively simple as

they can be mapped by a homeomorphism onto a family of parallel t r a j e c t o r i e s ,

see Fig. 1.1.

Fig. 1.1. Mapping of the paths in a regular domain into a field

of p a r a l l e l paths.

An i m p o r t a n t c l a s s of s o l u t i o n s of Eq. (1.1) are the so-called equilibrium or

steady solutions: X = X with X satisfying f(X ) = 0. An equilibrium or steady

s t a t e s o l u t i o n i s thus r e p r e s e n t e d in the phase space by a single point and

corresponds to a c r i t i c a l point of the v e c t o r f i e l d , in the following such a

point will often be referred to as a singular point of the d i f f e r e n t i a l equation

( 1 . 1 ) . Steady solutions: X= X cannot be reached by a neighbouring s o l u t i o n

X(t) i n f i n i t e time. A steady solution i s stable if a solution X(t) through a

point in a neighbourhood of X , remains close to t h a t steady solution for t + ».

The steady s o l u t i o n i s a s y m p t o t i c a l l y s t a b l e i f a l l neighbouring solutions

-7-X(t) ■» X as t ■» <». Steady solutions not satisfying the stability condition are said to be unstable.

Some examples of stable and unstable solutions in the phase space are depicted in Fig. 1.2.

a:stable b:asympfoficatty sfable ^unstable

Fig. 1.2. Stable and unstable steady solutions.

The phase portrait of trajectories near singular points can be rather compli­ cated and is usually not homeomorphic with a field of parallel trajectories. Important elements of phase portraits of systems are singular points and the local trajectory pattern near these points.

Therefore, a systematic description of phase portraits near singular points is a useful tool when analysing phase portraits of systems. A detailed treatment of phase patterns near singular points is given in paragraph 2 of this chapter.

Another class of solutions of Eq. (1.1) we want to mention here are the so-called periodic solutions satisfying: X(t+T) = X(t) with period T. Periodic solutions of a dynamical system appear as closed paths in the phase space. Closed paths in R2, representing periodic solutions of two-dimensional systems, must divide the phase plane into an inner and an outer region. If there exists a neighbourhood of a closed path which does not contain another closed path, this path is called a limit cycle. A limit cycle is stable if all neighbouring paths of the limit cycle approach it if t ■> <■>, otherwise the limit cycle is unstable: see Fig. 1.3.

An important question, but often difficult to answer, is to determine whether an autonomous system has periodic solutions and where they appear in the phase space.

-8-a:stable limit cycle b:unsfable limit cycle

F i g . 1 . 3 . Limit c y c l e i n R1

An e l e m e n t a r y c o n d i t i o n t h a t can be used f o r the n o n - e x i s t e n c e of c l o s e d p a t h s i s formulated by Bendixson's c r i t e r i o n .

B e n d i x s o n ' s c r i t e r i o n

Let x = P ( x , y ) and y = Q ( x , y ) be an a n a l y t i c a l d y n a m i c a l s y s t e m and l e t U be a s i m p l y - c o n n e c t e d domain of the phase p l a n e on which t h e d i v e r g e n c e of t h e v e c t o r f i e l d : d i v ( P . Q ) d o e s n o t change s i g n and i s not i d e n t i c a l l y z e r o . Then t h e r e a r e no c l o s e d p a t h s l y i n g e n t i r e l y i n U.

The proof of B e n d i x s o n ' s theorem goes a s f o l l o w s .

A p p l i c a t i o n of the d i v e r g e n c e theorem along a c l o s e d curve T l y i n g e n t i r e l y i n U g i v e s

;/

i£ * ^

dx dy

= '

( P

-

Q

> •

n ds

UY 3 X 8 y T

with U the interior of T, n the outward normal and ds a line element of T. If 7 is a path of the vector field (P,Q), then (P.Q) and n are perpendicular so that the line integral I vanishes identically. But since the integrand of the

T

integral II is of one sign, the integral cannot be zero. Then the curve T cannot UT

-9-2. Phase portraits near singular points

n

Assume t h a t a dynamical system X = f (X), X e R has a singular point X so that f(X ) = 0 . In order to characterize the t r a j e c t o r y p a t t e r n n e a r X we assume f(X) t o be s u f f i c i e n t l y smooth and we expand f(X) near X . Retaining only the l i n e a r p a r t in the expansion there follows with X - X = E

o

5 = Df(XQ) 6. 5 e Rn (1.2)

3f .

where Df(X ) denotes the Jacobian matrix [-—] of the f i r s t o r d e r p a r t i a l o o x .

J derivatives of the vector function

^ ( x j . . . . - xn) f2( xr . . . xn)

fn( xl '

• s » /

evaluated in the singular point X

Equation (1.2) is a linear system with constant coefficients and can be analysed with classical methods yielding the corresponding 'linearized' phase portraits. However these, 'linearized' phase portraits are not necessarily equivalent with those near X of the original non-linear system.

The relation between both phase portraits is given by the theorem of Hartman-Grobman which holds for systems in R .

Hartman-Grobman

If Df (X ) has no eigenvalues with zero real part, then the family of trajectories near a singular point X of a nonlinear system X =

o

f(X)* and those of the locally linearized system have the same topological structure; which means that in a neighbourhood of X there exists a homeomorphic mapping which maps trajectories of the non-linear system into trajectories of the linear system.

When Df (X ) has no eigenvalues with zero real part, the singular point X is called hyperbolic or nondegenerate.

* In this thesis we restrict ourselves to the application of cases where f is analytic but the theorem can be extended to a wider,class of functions.

-10-The theorem of Hartman-Grobman implies that local linearization near singular

points will be an effective method when analysing phase p o r t r a i t s of n o n - l i n e a r

systems. Therefore i t i s worthwhile to recall f i r s t the theory of linear systems

in some d e t a i l .

Linear systems

Linear systems in R are t r e a t e d e x t e n s i v e l y and in v a r i o u s a s p e c t s i n the

l i t e r a t u r e .

Especially, systems in R

2

are very well known and receive much attention in many

t e x t b o o k s on o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , e . g . Coddington and Levinson

(1955) and Jordan and Smith (1977).

Furthermore, for l i n e a r systems in R' reference may be made to Reyn (1964),

where a sound treatment of s i n g u l a r p o i n t s culminates i n t o a conveniently

arranged survey of three-dimensional phase p o r t r a i t s .

Let us b r i e f l y r e c a l l here the main r e s u l t s for l i n e a r systems in R

2

, as they

are frequently used in the following chapters.

Consider the two-dimensional system

where A i s a constant 2x2 matrix.

The system has only one singular point: the origin (0,0).

A c l a s s i f i c a t i o n of phase p o r t r a i t s can be g i v e n i n t e r m s of t h e t r a c e

(p = A.+A_) and Jacobian (q = A..A_) of A; A., A_ being the eigenvalues of A.

We consider several cases

(i) q < 0: the eigenvalues A. and A_ are real with different signs.

The singular point i s a saddle point and i s unstable.

( i i ) 0 < q < (j-p)

2

: the eigenvalues A. and A_ are r e a l , unequal and of the same

sign. The singular point i s a stable node (sink) i f p < 0 and an u n s t a b l e

node (source) i f p > 0.

( i i i ) q > (j-p)

2

(p * 0 ) : the eigenvalues A- and A_ are conjugate complex with

non zero r e a l p a r t . The phase p o r t r a i t shows a focus or s p i r a l near the

singular p o i n t . The s i n g u l a r p o i n t i s a s t a b l e focus i f p < 0 and an

unstable focus i f p > 0.

-11-(iv) q > O, p = O: the eigenvalues A and A? are purely imaginary. T h e trajec­ tories form closed curves surrounding the origin and the singular point is called a center,

(v) q = (S-p)' (p * 0 ) : the eigenvalues A. and A_ are real and equal (A.. = A_ = A ) . The nature of the phase portrait depends on the Jordan form of A. If A has the Jordan form (_ ) the trajectories form a star shaped node, which is stable for p < 0 and unstable if p > 0.

If A has the Jordan form L .) the phase portrait is called an inflected node; all trajectories tend to the origin in the same direction and are parallel at infinity.

(vi) q = 0, p * 0: one of the eigenvalues is zero; the phase portrait consists of a family of parallel paths and a line of singular points, which is stable for p < 0 and unstable if p > 0.

This classification of linear systems is conveniently arranged in the p-q-plane, shown in Fig. 1.4.

q = XvX

p = X, + \2

-12-The isolated singular points for a linear system are saddles, nodes, foci and centers.

Since for nodes, f o c i and s a d d l e p o i n t s Re(A) * 0 h o l d s , Hartman-Grobman's theorem implies t h a t adding s u f f i c i e n t l y smooth n o n - l i n e a r i t i e s does not change the phase p o r t r a i t s near these types of singular p o i n t s .

C e n t e r s i n n o n - l i n e a r systems s a t i s f y Re(A) = 0, and t h e i r existence cannot be shown by l i n e a r i z a t i o n . As an i l l u s t r a t i o n consider the nonlinear system

x = y

y = - x + G x2 y

with eigenvalues A , A_ = ± i. Unless e = 0, the singular point (0,0) is not a center as in the linearized system, but a nonlinear focus, stable if E < 0.

Nonhyperbolic singularities as characterized by a vanishing Jacobian (q = 0) can appear in the phase space in different forms: isolated points, curves etc. The isolated points are usually denoted as multiple-equilibrium points, degenerate singularities or higher-order singularities. The topological structure of the local phase portrait near higher-order singularities can be very complicated and a general classification for singularities in R is hard to give.

For R2 , Andronov et. al (1973) offer a classification for isolated higher-order singularities having a nonvanishing degenerate linear part. These points will be reviewed in more detail in paragraph k.

-13-3.Topological structure of phase portraits, structural stability, bifurcation

ï2E2ï2Sï2?ï Ë

truc

£yE§_2?_E!}&˧_P2E£ï&iE§

The concept of t o p o l o g i c a l s t r u c t u r e of phase p o r t r a i t s has a l r e a d y been

introduced in paragraph 1 when we mentioned the fundamental theorem of

Hartman-Grobman.

Now a more thorough examination of t h i s concept will be given because i t enables

us to introduce and explain the concept of s t r u c t u r a l s t a b i l i t y .

C h a r a c t e r i s t i c f e a t u r e s of the phase p o r t r a i t which may be called q u a l i t a t i v e

properties are for example the number and type of singular p o i n t s , the existence

of closed paths and regions of a t t r a c t i o n . Formally, one may define q u a l i t a t i v e

properties as those p r o p e r t i e s of the phase p o r t r a i t which remain i n v a r i a n t

under a topological mapping or homeomorphism. A topological mapping between two

regions in de plane i s a one-to-one and bicontinuous mapping, meaning t h a t each

p o i n t M i s mapped exactly onto one point M' and t h a t d i s t i n c t points M. and M

?

are mapped onto d i s t i n c t points M' and Mi, and the mapping i s continuous e i t h e r

way.

An i n t u i t i v e description of a topological mapping of the p l a n e onto i t s e l f i s

given by Andronov e t . al (1973) as follows:

'imagine the plane i s to be made from rubber which i s deformed in some

way, stretching and squeezing i t at various points, but without tearing

or folding. Any topological mapping of the plane into i t s e l f i s either a

deformation of the above type (without tearing and folding) or a

mirror-reflection of the plane followed by such a deformation'.

I t may be c l e a r t h a t a t o p o l o g i c a l mapping of a phase p o r t r a i t can r e s u l t in

d r a s t i c changes in the shape of t r a j e c t o r i e s , but c e r t a i n p r o p e r t i e s w i l l

n e v e r t h e l e s s be p r e s e r v e d . As an example: s i n g u l a r p o i n t s a r e mapped i n t o

singular points and closed curves remain closed curves.

The concept of topological structure of a phase p o r t r a i t in R

2

i s now indirectly

indicated by the following definition.

Consider in a region G d f f two dynamical systems D. and D_, given by

X = f^X) (D

1

)

-14-The phase portraits of the systems (D.) and (D_) have the same topological structure if there exists a topological mapping (homeomorphism): T which maps G onto G and which takes paths of (D.) over into paths of (D_)*).

If two points X. and X? lie on the same path of system D., then their images TX. and TX_ lie on the same path of system (D_).

Also, if two points X1 and X_ lie on the same path of system (D_), then their - 1 - 1

images T X. and T X_ lie on the same path of system (D.).

The given definition of topological structure is in a certain sense indirect since it does not state exactly what topological structure is, but it specifies the necessary conditions for equal topological structures.

Structural_stability; bifurcation

After the definition of equivalent topological structure, we now introduce the concept of structural stability.

Consider the dynamical system (S) defined in a region G C R .

X = f(X) (S)

System (S) is said to be structurally stable if an infinitesimal change of f(X) leaves the topological structure unaffected in G, otherwise the system is called structurally unstable.

It should be noticed, that structural stability is not an intrinsic property of a topological structure but is related to the class of infinitesimal changes of f (X) that are allowed. Both the class of perturbations of the vector field which are admitted and f(X) itself determine whether there is structural stability or not.

') If the topological mapping T (with inverse T ) is k times differentiable (k > 0 ) , the mapping is called C diffeomorph and the two vector fields are

k k

said to be C -equivalent; C -equivalence with k > 0 implies that certain smoothness properties (k-times differentiable) of trajectories remain preserved by the mapping process.

-15-This c h a r a c t e r i s t i c property of s t r u c t u r a l s t a b i l i t y i s i l l u s t r a t e d i n the next example where two d i f f e r e n t types of perturbations are imposed on the l i n e a r system (in R* ):

x = - x

(1.3) y = -2y

which has a node a t the o r i g i n .

F i r s t we c o n s i d e r an a n a l y t i c p e r t u r b a t i o n by e.^ing a l i n e a r term ux to the right-hand side of y:

x = - x

(1.4) y = -2y + ux

with u being a small perturbation parameter.

The origin is a stable node for u = 0 as well for u * 0, thus (1.3) is struc­ turally stable with respect to the perturbations of the (analytical) character given in Eq. (1.4).

Next we consider a non-analytic perturbation*) of (1.3) sy adding a perturbation of the form u /|x|

x = u /|x| - x

(1.5) y = -2y

If li > 0 system (1.5) has two singular points on the x-axis: (0,0) and (u2,0). Near (0,0) the trajectories behave partially like those near a stable node and partially like those near a saddle point. At (uJ,0) there occurs a stable node. For ii < 0 similar results will follow. The phase portraits obtained if u varies near zero are shown in Fig. 1,5.

*) Although such non-analytical perturbations seem a bit far-fetched, they appear in actual flow situations, as can be observed in Chapter II where structural stability in conical flows around an external corner is treated.

1 6 -M = 0 |j>0 F i g . 1.5. Nonanalytic p e r t u r b a t i o n of a nodal p o i n t . The e x a m p l e s i l l u s t r a t e t h a t t h e s t a b l e node a t ( 0 , 0 ) i s s t r u c t u r a l l y s t a b l e a g a i n s t a n a l y t i c a l p e r t u r b a t i o n s (Eq. ( 1 . 4 ) ) b u t s t r u c t u r a l l y u n s t a b l e a g a i n s t p e r t u r b a t i o n s ( n o n a n a l y t i c a l ) a s g i v e n by Eq. ( 1 . 5 ) . E v i d e n t l y , s t r u c t u r a l s t a b i l i t y of a t o p o l o g i c a l s t r u c t u r e cannot be e s t a b l i s h e d i n d e p e n d e n t of t h e c l a s s of p e r t u r b a t i o n s t h a t a r e imposed on t h e dynamical system.

To g a t h e r more a s p e c t s about s t r u c t u r a l s t a b i l i t y and l e a v i n g t h e d i s c u s s i o n s as s i m p l e a s p o s s i b l e , l e t u s c o n t i n u e by c o n s i d e r i n g systems i n R2 i n some more d e t a i l . Assume t h a t the l i n e a r system:

A X X 6 R2 (1.6)

is perturbed by adding a term u f(X) yielding the nonlinear system

X = A X + u f(X) u € R (1.7)

where f(X) at least Cl and f(0) not necessarily equal to zero.

If the eigenvalues A-, A_ of A have non-vanishing real parts, Re(A. _) * 0 then it can be shown that near the origin the phase portrait of (1.6) is structurally stable against the C1-perturbations given in Eq. (1.7). To this end, the phase portrait of Eq. (1.7) near the origin has to be determined. A singular point: X of Eq. (1.7) satisfies

A X + u f(X ) = 0

o K o'

Since A is an invertible matrix, and X = o

-u A - 1 f(X ) the implicit function t h e o r e m can be u s e d t o f i n d n e a r X

s u f f i c i e n t l y s m a l l p .

-17-The phase portrait of Eq. (1.7) follows by considering the locally linearized system near X , of which the coefficient matrix A + p Df(X ) has eigenvalues which depend continuously on the perturbation parameter p. For small p, compared with |Re(A. _ ) | , (Re(A .) * 0) the eigenvalues of A + p Df(X ) cannot cross the imaginary axis so that X is also a hyperbolic singularity. From Hartman-Grobman's theorem, then follows that the systems (1.6) and (1.7) have phase portraits near (0,0) which are topologically equivalent.

Hence, phase portraits near hyperbolic points are structurally stable with respect to C1-perturbations, phase portraits near nonhyperbolic points may loose their topological structure if C' -perturbations are imposed on the dynamical system. This phenomenon where a small variation of the system causes a change of topological structure is called a bifurcation. If the changes are caused by perturbations containing parameters, those parameters which actually cause bifurcation are called bifurcation parameters.

Two types of bifurcations in phase portraits may be distinguished: local^ bifurcations and global bifurcations.

The former appear if the topology of the phase portrait is only .locally affected. Such bifurcations can be observed with local analyses, they occur in particular if nonhyperbolic points are present; bifurcation changes the topology of the phase portrait only in a small neighbourhood of the nonhyperbolic singularity. In those cases where local analyses fail to detect bifurcation effects, the bifurcation is called a global bifurcation.

Saddle-connections and multiple limit cycles are well known examples of global bifurcations. Let us discuss them briefly for phase portraits in R* .

Saddle_connection

Consider a dynamical system in R* having a phase portrait which contains a special trajectory connecting two (hyperbolic) saddle points. Suppose that the system is perturbed such that the saddle connection breaks up resulting in its disappearance. Although the topology in the phase plane is significantly altered, see Fig. 1.6, the disappearance of the saddle connection cannot be detected by only a local examination of the phase portraits; to observe the broken saddle connection a broader view, containing the saddle points together with the separatrices, is necessary. The imposed perturbations have a so-called global effect on the trajectory pattern and as a result the transition process is called a global bifurcation.

1 8

-saddle connection broken saddle connection

F i g . 1.6. P e r t u r b a t i o n o f s a d d l e c o n n e c t i o n , g l o b a l b i f u r c a t i o n .

A n o t h e r e x a m p l e o f a g l o b a l b i f u r c a t i o n m a y a p p e a r i f a n isolated c l o s e d p a t h : T i s e m b e d d e d i n a t r a j e c t o r y p a t t e r n i n R' a s shown i n F i g . 1 . 7 - A s s u m e t h a t

f o r i n c r e a s i n g t t h e t r a j e c t o r i e s i n the o u t e r r e g i o n a r e s p i r a l l i n g i n w a r d s , all t e n d i n g to T i f t -» ». I n t h e i n n e r r e g i o n o f T they tend to T f o r t -> —•

a n d t o a f o c a l p o i n t f o r t ■» °>. T h e i s o l a t e d closed p a t h T i s a (semistable) limit c y c l e , w h i c h c a n n o t b e reached i n finite t i m e along trajectories s t a r t i n g i n a n e i g h b o u r h o o d o f T .

c

T h e p h a s e p o r t r a i t a s s k e t c h e d i n F i g . 1.7 can b e g e n e r a t e d b y the p a t h s o f t h e s y s t e m

r = - r ( r - l )2 , 9 = 1

w h e r e ( r , 8 ) a r e p o l a r c o o r d i n a t e s i n the p h a s e p l a n e . T h e closed p a t h T i s then found o n the c i r c l e r = 1.

u = 0

7^1

u>0

Fig. 1.7. Perturbation of limit cycle, global bifurcation.

Consider a perturbation of the system by adding the linear term ur (u 6 R) to the right-hand side of r; u appears to be a bifurcation parameter.

-19-The phase portrait of the perturbed system

r = -r(r-l)2 + ur, 9 = 1

has two closed paths: r. _ = 1 ±J~v if u < 0 and no closed path if p > 0. The appearance (disappearance) of closed paths has a global effect on the topologi-cal structure of the phase portraits which can not be established by only a local examination of the trajectory pattern.

The previous remarks point out that bifurcations - local or global - may appear if the phase portrait is structural unstable.

Nonhyperbolic singular points appear as elements in phase portraits that may cause local bifurcations; on the other hand saddle-connections and closed paths can give rise to bifurcations with a global character.

Whether a bifurcation actually occurs is not only determined by the presence of structural unstable elements in a phase portrait but depends also on the class of perturbations that is admitted.

The last statement applies also to global bifurcations, as can be shown by 3H 3H

perturbing a Hamiltonian system: x = — , y = - — such that the perturbed system remains Hamiltonian. Then closed trajectories can remain closed preventing a global bifurcation to appear.

-20-4. Higher-order singularities in R2

In p a r a g r a p h 3 we have seen t h a t h y p e r b o l i c and nonhyperbolic p o i n t s are

s t r u c t u r a l l y unstable with respect to an a p p r o p r i a t e c l a s s of p e r t u r b a t i o n s .

T r a j e c t o r y p a t t e r n s near nonhyperbolic p o i n t s can be very complex and in

general, as will be seen in the next paragraph, they will change i n t o p a t t e r n s

near combinations of hyperbolic singular points i f the system is perturbed.

From t h i s point of view one might argue that nonhyperbolic p o i n t s a r e f a r from

i n t e r e s t i n g because they are so ' e x o t i c ' that they are hardly met in practical

s i t u a t i o n s where small perturbations, always present, lead them desintegrate.

Obviously, a d i f f e r e n t view r e s u l t s if■ non-hyperbolic points can be helpful to

analyse t r a n s i t i o n processes in phase p o r t r a i t s , i . e . if an a p p r o p r i a t e change

of parameters in a dynamical system changes the topological structure.

In t h a t case, special combinations of parameter values e x i s t at which one or

more s i n g u l a r p o i n t s i n the phase p a t t e r n become nonhyperbolic. For such a

parameter combination the system i s called degenerate or nongeneric; however, a

small change of these parameter values can convert the nongeneric system into a

non-degenerate or generic one.

Such a d e g e n e r a t e s t a t e of the system marks a point in the parameter space at

which the topological properties of the system change qualitatively.

These o b s e r v a t i o n s give us sufficient motivation for a more detailed treatment

of these ' e x o t i c ' nonhyperbolic points.

The discussion will be r e s t r i c t e d to nonhyperbolic points in R

2

since only these

will be found to occur in the subsequent part of t h i s work.

The theory of nonhyperbolic singular points in R

2

i s well developed by Andronov

e t . a l (1973) and the following review of the most important results serves as a

b a s i s f o r applications. For a profound treatment including mathematical proofs,

we refer to Andronov e t . al (1973)•

Andronov c o n s i d e r s an i s o l a t e d nonhyperbolic s i n g u l a r p o i n t of an analytic

vector field in R

2

such that the expansion near the singular point involves a t

l e a s t one f i r s t - o r d e r term. Then a distinction can be made between s i n g u l a r i t i e s

having one or both eigenvalues, zero.

-21-Higher-order singular points with one zero eigenvalue; p * 0

Suppose that 0(0,0) is an isolated singular point of a planar system with one nonzero eigenvalue.

Then the system can be written as

x = P(x,y) = ax + by + P2(x,y)

y = Q(x,y) = ex + dy + Q2<x,y)

where P2(x,y) and Q_(x,y) are analytic with terms not lower than second degree and for the eigenvalues in 0(0,0) we have

p = A . + A?= a + d * 0

q = A.A = ad - be = 0

This system can be transformed by a nonsingular linear coordinate transformation in the canonical form

x = P(x,y) = P2(x,y)

y = Q(x,y) = y + Q2(x,y)

For isolated points i t i s assumed that P_(x,y) ^ 0.

For this system Andronov has shown t h a t t r a j e c t o r i e s e x i s t t h a t tend to the

s i n g u l a r p o i n t i n a d e f i n i t e d i r e c t i o n ( s e m i p a t h s ) . These semipaths tend to

0(0,0) only in the directions 0, p, n and ^ - ; only one semipath e x i s t s i n the

d i r e c t i o n -z and only one in the direction ~-; these two special semipaths are

denoted L

1

and L_ respectively.

To obtain the p o s s i b l e topological structures near the singular point consider

the equation y + Q

?

(x,y) = 0. By the implicit function theorem t h i s equation has

e x a c t l y one s o l u t i o n y = <p(x) i n a neighbourhood of 0 ( 0 , 0 ) , where <p(x) i s

analytic and obeys the condition <p(o) = <t>' (o) = 0. The curve y = <p(x) i s an

isocline of horizontal directions (y = 0) of the vector f i e l d .

The next step i s to define a function:

2 2

-which g i v e s t h e v a l u e of x on t h e c u r v e y = 0 . B e c a u s e 0 ( 0 , 0 ) i s i s o l a t e d (?2 ^ 0 ) , i|i(x) assumes t h e form

* ( x ) = A xm + o ( xm) , m > 2

m —

where A is the first nonvanishing coefficient (A * 0) and m is integer.

Let us consider the neighbourhood U. of 0(0,0) bounded by a circle C with radius <5, see fig. 1.8.

The curve y = <t>(x) d i v i d e s t h e domain U, i n t o two p a r t s . S i n c e y = y + Q2( x , y ) , y > 0 i n t h e upper r e g i o n , s o t h a t t h e v e c t o r f i e l d p o i n t s upwards t h e r e . S i m i l a r l y we have downwash i n t h e l o w e r r e g i o n . T h i s i m p l i e s t h a t t h e semipaths L. and L_, l y i n g above and below t h e c u r v e y = <p(x) r e s p e c t i v e l y , a r e u n ­ s t a b l e . Since <t>(x) i s a n a l y t i c , e i t h e r y = <p(x) c o i n c i d e s w i t h t h e x - a x i s (<p(x) = 0) or i t h a s , i n a s u f f i c i e n t l y s m a l l neighbourhood of 0 ( 0 , 0 ) , no o t h e r p o i n t s than 0 ( 0 , 0 ) i n common with the x -a x i s .

*P(x)

Fig. 1.8. Neighbourhood U- of 0(0,0) o

In the first case the positive and negative x-axes are obviously semipaths of the system, while in the second case (<p(x)^0) the trajectories must cross the curve y = <P(x). These trajectories are passed along in a direction depending on the sign of t|>(x).

With the expression i|i(x) _{A x} m

+ we observe that this depends on the sign A and on the parity of m, which implies that four different cases have to be considered (A > 0 or < 0, m is odd or even). According to Andronov, these considerations lead to a classification of topological structures which is summarized in Andronov's theorem 65 (p. 3^0, Andronov et. al,

(1973))-Theorem 65. Higher-order singular points with only one zero eigenvalue

Let 0(0,0) be an isolated singular point of the system x = P_(x,y) and y = y + Qp(x,y) where P_ and Q? are analytic and have series expansions near 0(0,0) of which the lowest-order terms are at least quadratic.

-23-Let y <p(x) be the solution of y + Q2(x,y) 0 in the vicinity of 0(0,0) and assume that the series expansion of the function *(x) = P2(x, <?(x)) has the form t|i(x) = A x + where m £ 2 and A * 0. Then:

m m 1. If m is odd and A > 0, 0(0,0) is a topological node with an infinite

number of semipaths in the directions 0,n and exactly one semipath tending to 0 in the direction -= and also one in the direction ~-,

2. If m is odd and A < 0, 0(0,0) is a topological saddle point whose

separa-m _

t r i c e s (semipaths) approach 0(0,0) i n t h e d i r e c t i o n 0 , -~, n and ~ - ,

respectively.

3 . If m i s even, 0(0,0) i s a saddle-node, i . e . a singular point whose n e i g h ­

bourhood c o n s i s t s of one p a r a b o l i c s e c t o r (nodal type) and two hyper­

bolic*) sectors (saddle-type). If A < 0 the hyperbolic sectors contain a

segment of the p o s i t i v e x - a x i s , if A > 0 they contain a segment of the

negative x-axis.

By definition the order of a singularity i s equal to m.

The full proof of t h i s theorem may be found in Andronov e t . a l (1973) PP- 337

ff. Typical phase plane p i c t u r e s near t h e s e multiple-equilibrium points are

shown in Fig. 1.9»

^i

< ■

k-topological node topological saddle saddle-node

Fig. 1.9. Higher-order s i n g u l a r i t i e s in R

2

with one zero eigenvalue.

Higher^order singular goints_having both_eigenvalues_zero£_g_=_0

In t h i s section we consider the system

x = ax + by + P

2

(x,y), y = ex + dy + Q

2

(x,y)

with the assumption p = a + d = 0 q = ad - be = 0

|a| + |b| + |c| + |d| * 0

*) Other examples of hyperbolic sectors appear near a hyperbolic saddle point where four hyperbolic sectors can be distinguished.

-2k-Suppose 0(0,0) is an isolated singular point at the origin and P,(x,y) and Q2(x,y) are analytic in the vicinity of 0. The series expansions of P (x,y) and Qp(x>y) involve terms not lower than second order. Then a nonsingular linear transformation exists (c.f. Andronov p. 3^7) bringing the system into the normal form

x = y + P2(x,y) y = Q2(x,y)

The topological structure of 0(0,0) is found by considering the isocline of vertical directions: y = <p(n

On this curve the functions

vertical directions: y = <t>(x) where <p(x) satisfies <p(x) + P-,(xt <p(x)) = 0.

3P-

3Q-*(x) = Q2(x, <p(x)) and p(x) = j ^ - (x, <f(x)) + j ^ (x, <p(x)J

may be evaluated and expanded as

i(i(x) = A x + , p(x) = b x +

m r' ' n

where A and b are the f i r s t nonvanishing coefficients in these expansions and m and n are integer.

The topological s t r u c t u r e of phase p o r t r a i t s in R2 near singular points with zero as a double eigenvalue is then established by Andronov's theorems 66 and 67

(PP. 356, 362).

Theorem 66, 67. Higher-order singular points having both eigenvalues zero Let 0(0,0) be an i s o l a t e d singular point of the system x = y + P_(x,y), y = Q2(x«y); l e t y = *(x) °e the solution of y + P?(x, y) = 0 near 0(0,0) and assume t h a t the s e r i e s expansions of the function 4> (x) = Qp(x, <t>(x))

3P2 3Q2

and p(x) = - — (x, <p(x)) + - — (x, <p(x)) have the respective forms ox oy

*(x) = A xm + ..., A * 0 m m p(x) = b x + . . . , b * 0 .

2 5

-1. Let the number m be odd, n = 2J + 1 (J 2 1) and A. = b2 + 4(8+1)4 .

Then i f A > 0 t h e s i n g u l a r p o i n t i s a t o p o l o g i c a l s a d d l e - p o i n t , b u t i f A < 0 t h e f o l l o w i n g p o s s i b i l i t i e s o c c u r :

- a focus o r a c e n t e r i f b = 0 ; i f b * 0, n > g; i f b * 0 , n = C, A < 0 n n n - a topological node if b * 0, n is even, n < 2; if b * 0, n is even,

n n n = e, A Ï 0

- a s i n g u l a r p o i n t w i t h an e l l i p t i c r e g i o n i f b * 0, n = odd, n < g; i f b * 0, n i s odd, n = 2, A è 0 .

2 . Let t h e number m be even, m = 22, (I £ 1 ) . Then the s i n g u l a r p o i n t i s

- a saddle-node i f b * 0, n < £ n

- a cusp if b = 0; if b * 0, n i £. n n

The order of the singularity is by definition equal to m.

The proof of this theorem may be found in Andronov et. al (1973) PP- 365 ff-Sketches of the phase portraits are shown in Fig. 1.10.

2 6 -m = odd = 2l+1

'

bn*0<

<

'

Am>0 bn=0 n=even n=odc ,X<0 X = b £ + 4 Am( H / ) ' X>0

)

topological saddle

y

^■¥^

focus, center topological node elliptic and hyperbolic sector

c

cusp saddle-node x=y + P2( x , y ) , y = Q2( x , y ) y|x=0 = 4 J ( x , = Amx m ÖP,

da

èx óy x=0 p ( x ) r bnXn -F i g . 1.10. H i g h e r - o r d e r s i n g u l a r i t i e s i n R2 w i t h two e i g e n v a l u e s z e r o .

-27"

5. Bifurcation of vector fields, unfoldings

In t h i s paragraph we study local b i f u r c a t i o n s as they occur i n v e c t o r f i e l d s n e a r s t r u c

defined as

n e a r s t r u c t u r a l l y u n s t a b l e s i n g u l a r p o i n t s . Consider a v e c t o r f i e l d in R

X = f (X), X 6 Rn, u e Rk

which depends on the k-dimensional parameter u = (p , u_, . . . . u, ) . The v e c t o r f i e l d f (X) i s assumed to be a n a l y t i c .

The term bifurcation introduced in paragraph 3 was o r i g i n a l l y used by Poincare t o d e s c r i b e t h e s p l i t t i n g behaviour of s t a t i o n a r y s o l u t i o n s : X = X of the dynamical system X = f (X). These solutions, which are r e p r e s e n t e d as s i n g u l a r p o i n t s in the phase s p a c e , can be found by s o l v i n g f (X ) = 0 for X as a

p o o function of u i f the Jacobian Df (X ) has no zero e i g e n v a l u e s . However, i f the

u o

Jacobian has a zero eigenvalue, occurring at some u, say u , the system X = f (X) is structurally unstable and several branches of the solution X = X (u)

n+k can come together at (X ,u ) in R

The parameter set u where the system looses its structural stability is called: bifurcation set, it can be viewed as dividing surfaces bordering domains in the parameter space where the system is non-degenerate (generic).

Variations in the parameter space intersecting the set (u|u = u } cause a change of the topological structure of the phase portrait.

Such changes are called bifurcations and the corresponding parameter values are the bifurcation values.

A one-parameter family of systems with k-1 relations between the parameters u. , u_, .... u, is represented at a curve A in the k-dimensional parameter space. Assume that A intersects the bifurcation set, u , and that the intersection is

c

transversal. At the intersection, structural unstability occurs, but due to the transversality condition the degeneracy can be removed by any small displacement along A. The corresponding perturbation is called a generic perturbation. Variations in the parameter space that coincide with u correspond to a non-generic perturbation.

-28-In this paragraph we focus the attention on bifurcations of isolated singular points. As we have seen these bifurcations are referred to as local bifurca­ tions , so that the vector field is studied near degenerate singular points and the bifurcating solutions are also found in the neighbourhood of these points. Let us start with a simple example of a dynamical system in R1 . Consider the one-dimensional 'vector' field:

x = f u^ = "* " x'

which depends on the scalar parameter u.

(1.8)

Here Df (x)

P 3x2, and the only bifurcation point is (x ,u (0,0). It is 0) and three dif-easy to check that f (x) = 0 has one solution if u i 0 (x

ferent solutions if u > 0 (x = 0 , ±-/p~) . A qualitative picture of these solu­ tions is given in Fig. 1.11a which shows the branches of singular points in a (x,u) space. This figure is called the bifurcation diagram; it shows the loci of singular points as a function of the parameters.

The phase space is one-dimensional and coincides with the x-axis. The point x = 0 is stable for p S 0 and it becomes unstable for u > 0.

The phase portraits which can occur after bifurcation are shown in Fig. 1.11b.

bifurcation point (b) (jïO: U>0:

-/Ji

_*

Fig. 1.11. Bifurcation of f (x) = ux - x3. (a) bifurcation diagram, (b) phase portraits.

The previous example of a parameter bifurcation gives the result of a one-parameter variation on the structurally unstable 'vector' field x = -x3.

The perturbed system x = +ux - x' shows some possible bifurcations and is called an unfolding of the unperturbed system x = -x3 . Now the important question arises: is the unfolding given by eq. (1.8) a particular one, showing only some bifurcations, or has the unfolding a more general character so that all possible bifurcations of x = -x', within a certain class of perturbations (for example

-29-analytic), are described. If the unfolding describes all possible bifurcations of the degenerate singular point, then it is called a universal unfolding. Hence a universal unfolding of a degenerate vector field is a family of bifurcating solutions which contains the bifurcation in a persistent way. It has the important property that it describes the bifurcation with the smallest number of parameters. This number is called the codimension of the degenerate vector field. The codimension of a bifurcation is the smallest dimension of the parameter space which contains the bifurcation in all its aspects. A universal unfolding is not necessarily unique since a coordinate transformation can lead to a differently formulated unfolding having the same dimension. Therefore the term 'universal unfolding' is not generally accepted in the literature; for example Shirer and Wells (1983) prefer the term versal unfolding to point out the nonuniqueness of the unfolding.

Bearing this in mind we return to the system x = -x' in order to find the universal unfolding of this system. Eq. (1.8) gives an unfolding with one parameter, but does it describe the bifurcation in all its aspects or are more parameters necessary for a universal unfolding?

If the function f(x) = -x1 is subjected to a small perturbation f(x) ■» f (x), it is obvious that f (x) possesses one, two or three zeroes near x = 0.

However, the unfolding (1.8) gives either one (u < 0) or three (u > 0) zeroes, which indicates that (1.8) is probably not a universal unfolding of x = -x1 . Because it is not possible to introduce more than three zeroes locally, all these behaviours can be captured by the addition of the lower-order terms u- + u_x, so that a universal unfolding of x = -x' is represented by the two-parameter family

fu(x) = jix + u2x - x' *) (1.9)

Equating f (x) to zero gives the loci of singular points x as a function of the parameters u. and u_. Figure 1.12 shows the corresponding bifurcation diagram with these loci in the x ,p1,u_ space.

h) Other universal unfoldings such as f (x) = u.x + u2x2 - x* or f (x) = u. + u_x2 - x' have the same bifurcation characteristics: they can be transformed into eq. (1.9) by a suitable translation of the x-coordinate.

-30-A s i n g u l a r p o i n t becomes degenerate i f

Df(x ) h a s e i g e n v a l u e s with zero r e a l

part, thus i f u_ - 3x

z

= 0.

Eliminating x from t h i s equation and the

equation f (x ) = 0 we find the parameter

combinations a t which b i f u r c a t i o n takes

place.

These parameter combinations are called

the bifurcation set which s a t i s f i e s :

Fig. 1.12. Bifurcation diagram of

f (x)

_y

_l

_{+ P}

₂

_X

_"

0

This s e t c o n s i s t s of two branches if u_ > 0, the corresponding one-dimensional

phase p o r t r a i t s are shown in Figure 1.13. We observe from the phase p o r t r a i t s

that on a branch (u_ > 0) the nonhyperbolic point i s the one-dimensional variant

of a saddle-node. Crossing a branch transversally implies the bifurcation of the

saddle-node. Such a crossing i s in fact described by one parameter so that the

branches represent a codimension-one bifurcation. At the p o i n t (u..,u_) = (0,0)

where both branches terminate with a common tangent we have a nonhyperbolic

one-dimensional node; i t s bifurcation i s described by two parameters implying t h a t

the point (u.. ,u_) = (0,0) represents a codimension-two bifurcation.

This example of a codimension-two bifur­

cation i l l u s t r a t e s that the unfolding of

x = - x ' given by eq. (1.9) c o n t a i n s ,

besides the generic bifurcation (already

given by eq. ( 1 . 8 ) ) , also n o n g e n e r i c

b i f u r c a t i o n s . The p o s s i b i l i t y to d e ­

scribe a l s o t h e s e nongeneric b i f u r c a ­

tions i s realized by introducing the two

parameter unfolding: f (x) = u

1

+y-x-x* ,

implying that the universal unfolding of

x = -x' i s defined with two parameters

involving a codimension-two bifurcation.

I f only g e n e r i c b i f u r c a t i o n s a r e of

i n t e r e s t i t i s enough to consider the

Fig. 1.13. Bifurcation set and phase one-parameter unfolding as given by eq.

p o r t r a i t s of x=u- +p_x-x' . (1.8).

{#-(#"

-31-The preceding consideration about bifurcations of a one-dimensional system illustrates that concepts like unfolding and codimension can be valuable tools in bifurcation analyses. Especially if the aim is to develop a general theory about degeneracies and their universal unfoldings, the concept codimension plays an important role. It offers the possibility to develop a strategy for the study of these degeneracies whereby one starts by investigating codimension-one bifur­ cation, then codimension-two bifurcations etc. Such a strategy can ultimately lead to a classification scheme of bifurcations based on general considerations such as number of parameters, dimension of the phase space and constraints which account for the class of perturbations, see for example Shirer and Wells (1983). In the next table we have summarized a few examples of elementary universal unfoldings of degenerate singularities occurring in one- and two-dimensional systems.

One-dimensional universal unfoldings

name fold cusp swallowtail x5 butterfly name

hyperbolic-umbilic

e l l i p t i c - u m b i l i c

As we have mentioned before, the presented form of the unfoldings i s not neces­

s a r i l y unique; s e v e r a l a l t e r a t i o n s can be found which account for the same

b i f u r c a t i o n behaviour. These alterations follow from suitable coordinate trans­

formations and rearrangement of the parameters.

The l o c i of the nearby s i n g u l a r i t i e s : x (p , p p . ) , forming the bifurca­

tion diagram can be portrayed as s u r f a c e s (boundaries) i n the x - p-space

(p e R ) . The complex geometrical structure of these surfaces near the origin

form codimension unfolding

1

U1 + u2x

u?x + p x< Px + p2x + p3xJ + p^x3

Two-dimensional universal unfoldings form codimension unfolding

x y x2 - y2 x' P j + p,,y + xy U2 + p . y + x2 - j | px + p3x + p^y + P2 + y2

/

-32-can be associated with well known elementary s i n g u l a r i t i e s appearing i n c a t a s ­

t r o p h e t h e o r y . This correspondence with elementary catastrophes i s reflected by

the names l i s t e d in the table.

-33-6. Center manifolds

Before proceeding with the applications in fluid flow problems, there is a general technique which has to be discussed first, as it can simplify the analysis of bifurcation problems considerably. This technique has the effect of introducing coordinate systems in which computations are more easily carried out. After using it one is left with a reduced system of differential equations containing all the qualitative features of the bifurcation. It must be empha­ sized that this technique, called center manifold theory, has a local character and is only applicable to bifurcations of singular points.

The center manifold theory provides a means for systematically reducing the dimension of the state spaces which need to be considered when analyzing bifur­ cations of a given type.

Suppose we have a system of nonlinear ordinary differential equations:

X = f(X) X e Rn

for which f(0) = 0.

The linearized system at X = 0 may be written as

X = Df(0).X

If Df(0) has no eigenvalues with zero r e a l p a r t , then Hartman-Grobman's theorem s t a t e s t h a t t h e e i g e n v a l u e s , with p o s i t i v e and negative r e a l p a r t s , determine the local phase p o r t r a i t near X = 0. I f t h e r e are e i g e n v a l u e s with zero r e a l p a r t s the t o p o l o g i c a l s t r u c t u r e can be quite complicated. Some examples in R* have already been given in t h i s chapter (paragraph ^ ) .

A s o l u t i o n of t h e l i n e a r i z e d s y s t e m , s a t i s f y i n g t h e i n i t i a l c o n d i t i o n X(t ) = X , X e R i s the vector valued function

v o o o X(t;Xo) = exp(t Df(0)) XQ A general s o l u t i o n of X = Df(0) X i s o b t a i n e d by l i n e a r s u p e r p o s i t i o n of n l i n e a r l y independent s o l u t i o n s : ( V f t ) , — v ( t ) } n 1 X(t) = X C. VJ( t ) j = l J

3 * »

-where C. (j = 1, ... n) is a constant.

A set of linearly independent solutions can be obtained from the real and imagi­ nary parts of the vector valued functions exp(A.t) XJ where X"' are the (generalized) eigenvectors associated with the eigenvalues (real or complex) Aj(j = 1, ... n ) .

Various subspaces spanned by (generalized) eigenvectors and filled with solution curves may be distinguished.

n the stable subspace, E : spanned by X' Xs

n the unstable subspace, E : spanned by X1 ... X

n

c c the center subspace, E : spanned by X' ... X

n

s •

where X1 , . . . X are n (generalized) eigenvectors of which the eigenvalues s s s

have negative real parts, n

X1 , ... X are n (generalized) eigenvectors of which the eigenvalues have positive real parts and

n

X1 , ... X are n (generalized) eigenvectors with eigenvalues having zero real parts.

Since the dimension of the whole phase space is n we have n + n + n = n. s u e

The adjectives s t a b l e , unstable and center r e f l e c t the behaviour of solutions in t h e i r eigenspaces respectively.

Solutions l y i n g i n E a r e c h a r a c t e r i z e d by e x p o n e n t i a l decay (monotonie o r

u c o s c i l l a t o r y ) , t h o s e lying in E by exponential growth and those lying in E by

n e i t h e r . In Fig. 1.14 we show some examples of invariant subspaces in R2 and R' . s u For nonlinear systems one can define the invariant subspaces W and W being the

s u s u nonlinear analogues of E and E r e s p e c t i v e l y . The subspaces W and W a r e

g frequently denoted as stable and unstable manifolds, they are tangent to E and Eu at the singular point where f(X) = 0.

The existence of these manifolds for nonlinear systems is established by the following theorem.

-35-Df(0)=(2 0l Eu = span{(1,OI,<0,1)} (b)

Df(0) = (J.»)

Eu = (1,0) Es = (0,1) Ec = 0

c o

D f ( o ,

=(-

,

o:.?)

EU=0,ES=(O.O,1) Ec = span{(1,0,0),(0,1,0)}

Fig. 1.14. Subspaces of linear flows.

Stable Manifold Theorem for hyperbolic singular points

S u p p o s e t h a t X = f (X) h a s a h y p e r b o l i c singular point X , then there exist stable and unstable manifolds W (X ) , W (X ) , o f the same dimensions n , n as o' o s u those of the eigenspaces E , E of the linearized system; at X , W (X ) is tangent to Es and WU(X ) is tangent to Eu; WS(X ) and WU(X ) are as smooth as the function f.

The proof of this theorem has been given by Hartman (1964) and more recently by Carr (1981).

We note that this theorem says nothing about the existence of center manifolds

c r W for nonlinear systems. For the case that the nonlinear system is C the

existence of center manifolds is established in the center manifold theorem which includes the results of the stable manifold theorem. The first proof of the center manifold theorem has been given by Kelley (1967).

3 6

-Center Manifold Theorem

L e t f be a C v e c t o r f i e l d on R v a n i s h i n g a t X = X so t h a t f(X ) = 0 . o o The s p e c t r u m of e i g e n v a l u e s A- A of Df(X ) i s d i v i d e d i n t o t h r e e p a r t s , a , o , a with s u c A e a i f Re A < 0 s A € a i f Re A = 0 c A 6 a i f Re A > 0 _u

The c o r r e s p o n d i n g g e n e r a l i z e d eigenspaces a r e E , E and Eu r e s p e c t i v e l y . Then t h e r e e x i s t C s t a b l e and u n s t a b l e manifolds W and W t a n g e n t t o E and E a t X ; r e s p e c t i v e l y and a C c e n t e r m a n i f o l d W t a n g e n t t o E a t X . The

s u c

manifolds W , W and W a r e each f i l l e d with s o l u t i o n c u r v e s .

The c e n t e r manifold theorem i m p l i e s t h a t a n o n l i n e a r s y s t e m f (X) c a n be s p l i t

s+u c "

u p , n e a r a s i n g u l a r p o i n t X , i n t o two s u b s y s t e m s f and f of which the l i n e a r p a r t s h a v e e i g e n v a l u e s w i t h n o n z e r o a n d w i t h z e r o r e a l p a r t s , r e s p e c t i v e l y .

Then one may w r i t e t h e system X = f(X) near t h e s i n g u l a r p o i n t X : (X ,Y ) f o r ­ mally as

X = fC(X,Y) = DfC(X ,Y ).(X-X ) + F(X,Y) o o o Y = fS + U(X,Y) = DfS+U(X ,Y ).(Y-Y ) + G(X,Y)

0 - 0 O

(1.10)

n n +n c s u

where X € R , Y e R and F, G are nonlinear higher-order terms which vanish at (Xo.Yo).

S i n c e t h e c e n t e r m a n i f o l d i s t a n g e n t t o E a t (X ,Y ) i t can be w r i t t e n ex-p l i c i t e l y

Y-Y = h(X) with h(X ) = Dh(X ) = 0 o o o

The c e n t e r manifold W may be viewed a s a p a r t i c u l a r s e t of s o l u t i o n s o f t h e o r i g i n a l s y s t e m X=f (X) , s a t i s f y i n g t h e a d d i t i o n a l c o n s t r a i n t s f(X )=Df(X )=0.